Alternatively, the identity of the two currents can be shown by direct calculation. The condensate current can be obtained by simple Galilean invariance argument, as discussed by Bardeen and Johnson [1]}. The vacuum current was first calculated by Ishii [2]}. We comment several aspects of his calculation. Firstly, he considered an unshifted superconducting gap, which means that his calculation directly addresses the vacuum current rather than the condensate current. Secondly, he considers a normal wire with many conducting channels, which appears as integration over transverse momentum. This is not a limitation since his results can trivially be generalized to arbitrary number of conducting channels by replacing the integral with a summation over channels with a proper coefficient [3]}. An important result Ishii gets is the expression (3.5), which gives the current expressed as sum over the imaginary (Matsubara) frequencies. While he gets this using t-matrix for Andreev scattering, an alternative way to obtain the same result is to use the quasiclassical approach by solving the Eilenberger equation at imaginary frequencies. It is also worth pointing out that this result is valid for any temperature and for any length \(L\)  of the normal region. Evaluation of the expression at \(T=0\)  for a long junction gives the vacuum current that is linear in \(\phi \)  in the range \(|\phi |<\pi \)  and agrees also in magnitude with the condensate current. Alternative approximate expressions of the current is given in Ref. [4]}.
